Conventional methods of performing spectral analysis of signals assume that the input signals have constant harmonic content during the sampling operation during an observation window. In many applications, signals are time-dependent (non-stationary or transient), and may be considered as stationary for purposes of observation only when the duration or length of the observation window is appropriately short. For example, the length of the observation window should be less than the period of intermittence or transience, so that the signal statistics remain stationary throughout the observation window. In contrast, in order to obtain good spectral resolution of the signal, the number of samples is increased, which requires use of a longer observation window.
Generally, a fast Fourier transform (FFT) is not suitable when a narrow frequency band of a non-stationary signal has to be analyzed because good spectral resolution is not possible with a short observation window. To improve both spectral resolution and resolution by using the FFT, a longer observation window, i.e., a large number of points, is required. However, a non-stationary signal cannot be considered stationary throughout a long duration observation window because non-stationary signals are not constant. Therefore, it is difficult to obtain good spectral resolution, while at the same time reducing the length of the observation window to accommodate a non-stationary signal. In other words, it may not be possible to increase the length of the observation window because the non-stationary signal can be assumed to be stationary only for a short time interval.
Another limitation of the FFT is that it requires analysis of the entire spectrum, even when evaluating the signal spectral content only in a limited frequency band. Accordingly, since a fixed resolution for the frequency band under consideration is needed, the entire spectrum must be analyzed with the same resolution. This may result in a long observation window, which prevents the assumption that the signal is stationary, as discussed above.
In comparison, a chirp z-transform (CZT) algorithm does not require analysis of the entire spectrum with the same resolution, as does the FFT. In this way, the CZT provides a limited spectral analysis with better spectral resolution since the CZT does not depend only on the length of the observation window, like the FFT (and other classic methods, such as Welch). Rather, the CZT depends on the ratio between the analyzed spectral band and sampling frequency. In this case, after choosing a frequency band, it is possible to obtain a good spectral resolution even with a considerably lower number of samples and consequently a shorter observation window. In this way the CZT allows all the performance parameters to be optimized even if a shorter observation window is required.
The CZT enables use of only the sampled data to obtain a zoomed signal display without any further operation to correct leakage or amplitude errors. Also, it is possible to use the CZT to reconstruct the total signal spectrum, even for non-stationary signals, by processing different CZTs on limited spectrum portions, using the same buffered sampled data. CZT alone does not provide an enhanced resolution spectrum for the zoomed or the entire bandwidth of the signal. Therefore, there is a need to obtain more accurate signal analysis by improving spectral resolution, which cannot be done using CZT alone.